Finite Group Databasebeta

← C3×#24631 #10643238

Subgroups — C3×#24631 #10643238

Order 768.

974 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#459C211yesmaximal, characteristic
2C2×C3×C4×SD16C211yesmaximal, characteristic, complemented
3C3×#2548C211yesmaximal, characteristic, complemented
4C3×#2354C211yesmaximal, characteristic, complemented
5C3×#2054C212yesmaximal, complemented
6C3×#2012C212yesmaximal, complemented
7C3×#1983C212yesmaximal, complemented
8C3×#1957C211yesmaximal, characteristic, complemented
9C3×#1939C211yesmaximal, characteristic, complemented
10C3×#1926C212yesmaximal, complemented
11C3×#1803C211yesmaximal, characteristic, complemented
12#24631C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×SD16C2211yescharacteristic
14C22×C3×D8C2212yes—
15C2×C3×#127C2211yescharacteristic
16C2×C3×#124C2212yes—
17C2×C3×C4×Q8C2211yescharacteristic
18C2×C3×C4×D4C2211yescharacteristic
19C2×C3×#117—21no—
20C3×#459—24no—
21C3×C4×SD16—24no—
22C2×C3×#106C2211yescharacteristic
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×C4⋊C8C2212yes—
25C2×C3×#103C2211yescharacteristic
26C2×C3×#103C2212yes—
27C2×C3×#102C2212yes—
28C2×C3×#102C2211yescharacteristic
29C2×C3×#102C2212yes—
30C2×C3×#102C2212yes—
31C2×C3×C4×C8C2211yescharacteristic
32C3×#394—22no—
33C3×#390—21no—
34C3×#390—21no—
35C3×#390C2212yes—
36C3×#390C2212yes—
37C3×#388—22no—
38C3×#384C2211yescharacteristic
39C3×#380—21no—
40C3×C4×C4⋊C4C2212yes—
41C3×C4×C4⋊C4C2211yescharacteristic
42C3×#340C2212yes—
43C3×#336C2212yes—
44C2×#459C2×C311yescharacteristic
45C2×C4×SD16C2×C311yescharacteristic, complemented
46#2548C2×C311yescharacteristic, complemented
47#2354C2×C311yescharacteristic, complemented
48#2054C2×C312yescomplemented
49#2012C2×C312yescomplemented
50#1983C2×C312yescomplemented
51#1957C2×C311yescharacteristic, complemented
52#1939C2×C311yescharacteristic, complemented
53#1926C2×C312yescomplemented
54#1803C2×C311yescharacteristic, complemented
55C22×C3×Q8C2311yescharacteristic
56C22×C3×D4—21no—
57C22×C3×D4—21no—
58C22×C3×D4C2312yes—
59C22×C3×D4C2311yescharacteristic
60C23×C3×C4—21no—
61C2×C3×SD16—22no—
62C2×C3×SD16—22no—
63C2×C3×SD16—22no—
64C2×C3×D8—24no—
65C2×C3×D8—42no—
66C2×C3×D8—42no—
67C22×C3×C8C2312yes—
68C22×C3×C8C2311yescharacteristic
69C22×C3×C8C2311yescharacteristic
70C3×#127—22no—
71C3×#127—22no—
72C3×#124—24no—
73C3×#124—42no—
74C3×C4×Q8—22no—
75C3×C4×Q8—22no—
76C3×C4×D4—22no—
77C3×C4×D4D414yes—
78C3×#117—44no—
79C2×C3×C4⋊C4—21no—
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4D411yescharacteristic
83C2×C3×C4⋊C4D411yescharacteristic
84C2×C3×C4⋊C4C2312yes—
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C22⋊C4—21no—
88C2×C3×C22⋊C4—22no—
89C2×C3×C22⋊C4—22no—
90C2×C3×C22⋊C4—22no—
91C2×C3×C22⋊C4—22no—
92C2×C3×C22⋊C4—22no—
93C2×C3×C22⋊C4—22no—
94C2×C3×C42—22no—
95C2×C3×C42—21no—
96C2×C3×C42C2312yes—
97C2×C3×C42C2311yescharacteristic
98C3×#106—22no—
99C3×C4⋊C8—22no—
100C3×C4⋊C8—42no—
101C3×#103—22no—
102C3×#103—42no—
103C3×#102—22no—
104C3×#102—24no—
105C3×#102—24no—
106C3×#102—42no—
107C3×C4×C8—22no—
108C3×#95—22no—
109C3×#95D412yes—
110C22×SD16C22×C311yescharacteristic
111C22×D8C22×C312yes—
112C2×#127C22×C311yescharacteristic
113C2×#124C22×C312yes—
114C2×C4×Q8C22×C311yescharacteristic
115C2×C4×D4C22×C311yescharacteristic
116C2×#117—21no—
117#459—24no—
118C4×SD16—24no—
119C2×#106C22×C311yescharacteristic
120C2×C4⋊C8C22×C311yescharacteristic
121C2×C4⋊C8C22×C312yes—
122C2×#103C22×C311yescharacteristic
123C2×#103C22×C312yes—
124C2×#102C22×C311yescharacteristic
125C2×#102C22×C312yes—
126C2×#102C22×C312yes—
127C2×#102C22×C312yes—
128C2×C4×C8C22×C311yescharacteristic
129#394—22no—
130#390—21no—
131#390—21no—
132#390C22×C312yes—
133#390C22×C312yes—
134#388—22no—
135#384C22×C311yescharacteristic
136#380—21no—
137C4×C4⋊C4C22×C312yes—
138C4×C4⋊C4C22×C311yescharacteristic
139#340C22×C312yes—
140#336C22×C312yes—
141C24×C3—21no—
142C24×C3—22no—
143C2×C3×Q8—21no—
144C2×C3×Q8—21no—
145C2×C3×Q8—21no—
146C2×C3×Q8—21no—
147C2×C3×Q8—21no—
148C2×C3×Q8#4112yes—
149C2×C3×D4—22no—
150C2×C3×D4—21no—
151C2×C3×D4—21no—
152C2×C3×D4—21no—
153C2×C3×D4—22no—
154C2×C3×D4—22no—
155C2×C3×D4—22no—
156C2×C3×D4—21no—
157C2×C3×D4—21no—
158C2×C3×D4—22no—
159C2×C3×D4—22no—
160C2×C3×D4—42no—
161C2×C3×D4—42no—
162C2×C3×D4—42no—
163C2×C3×D4—42no—
164C2×C3×D4—81no—
165C2×C3×D4—81no—
166C2×C3×D4C2×D412yes—
167C22×C3×C4—21no—
168C22×C3×C4—21no—
169C22×C3×C4—24no—
170C22×C3×C4—22no—
171C22×C3×C4—21no—
172C22×C3×C4—42no—
173C22×C3×C4—42no—
174C22×C3×C4C2×D412yes—
175C22×C3×C4C2×D412yes—
176C22×C3×C4C2×D411yescharacteristic
177C22×C3×C4C2×D411yescharacteristic
178C22×C3×C4C2411yescharacteristic
179C3×SD16—28no—
180C3×D8—48no—
181C2×C3×C8—21no—
182C2×C3×C8—21no—
183C2×C3×C8—21no—
184C2×C3×C8—22no—
185C2×C3×C8—21no—
186C2×C3×C8—22no—
187C2×C3×C8—21no—
188C2×C3×C8—22no—
189C2×C3×C8#4112yes—
190C3×C4⋊C4—22no—
191C3×C4⋊C4—22no—
192C3×C4⋊C4—21no—
193C3×C4⋊C4—21no—
194C3×C4⋊C4—22no—
195C3×C4⋊C4—21no—
196C3×C4⋊C4—22no—
197C3×C4⋊C4—42no—
198C3×C4⋊C4—42no—
199C3×C4⋊C4—81no—
200C3×C4⋊C4C2×D412yes—
201C3×C22⋊C4—24no—
202C3×C22⋊C4—44no—
203C3×C22⋊C4—44no—
204C3×C22⋊C4—44no—
205C3×C22⋊C4—82no—
206C3×C22⋊C4—82no—
207C3×C22⋊C4—82no—
208C3×C42—22no—
209C3×C42—22no—
210C3×C42—44no—
211C3×C42—42no—
212C3×C42C2×D412yes—
213C3×C42#4112yes—
214C22×Q8C23×C311yescharacteristic
215C22×D4—21no—
216C22×D4—21no—
217C22×D4C23×C312yes—
218C22×D4C23×C311yescharacteristic
219C23×C4—21no—
220C2×SD16—22no—
221C2×SD16—22no—
222C2×SD16—22no—
223C2×D8—24no—
224C2×D8—42no—
225C2×D8—42no—
226C22×C8C23×C312yes—
227C22×C8C23×C311yescharacteristic
228C22×C8C23×C311yescharacteristic
229#127—22no—
230#127—22no—
231#124—24no—
232#124—42no—
233C4×Q8—22no—
234C4×Q8—22no—
235C4×D4—22no—
236C4×D4C3×D414yes—
237#117—44no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4C3×D411yescharacteristic
242C2×C4⋊C4C3×D411yescharacteristic
243C2×C4⋊C4C23×C311yescharacteristic
244C2×C4⋊C4C23×C312yes—
245C2×C4⋊C4C23×C311yescharacteristic
246C2×C22⋊C4—21no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C42—22no—
254C2×C42—21no—
255C2×C42C23×C312yes—
256C2×C42C23×C311yescharacteristic
257#106—22no—
258C4⋊C8—22no—
259C4⋊C8—42no—
260#103—22no—
261#103—42no—
262#102—22no—
263#102—24no—
264#102—24no—
265#102—42no—
266C4×C8—22no—
267#95—22no—
268#95C3×D412yes—
269C23×C3—22no—
270C23×C3—41no—
271C23×C3—41no—
272C23×C3—41no—
273C23×C3—41no—
274C23×C3—41no—
275C23×C3—41no—
276C23×C3—42no—
277C23×C3—42no—
278C23×C3—42no—
279C23×C3—42no—
280C23×C3—42no—
281C23×C3—42no—
282C23×C3—42no—
283C23×C3C22×D411yescharacteristic
284C3×Q8—22no—
285C3×Q8—22no—
286C3×Q8—41no—
287C3×Q8—41no—
288C3×D4—22no—
289C3×D4—22no—
290C3×D4—42no—
291C3×D4—42no—
292C3×D4—41no—
293C3×D4—42no—
294C3×D4—41no—
295C3×D4—42no—
296C3×D4—84no—
297C3×D4—84no—
298C2×C3×C4—21no—
299C2×C3×C4—21no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—22no—
308C2×C3×C4—41no—
309C2×C3×C4—41no—
310C2×C3×C4—41no—
311C2×C3×C4—41no—
312C2×C3×C4—48no—
313C2×C3×C4—41no—
314C2×C3×C4—42no—
315C2×C3×C4—42no—
316C2×C3×C4—41no—
317C2×C3×C4—42no—
318C2×C3×C4—41no—
319C2×C3×C4#12112yes—
320C2×C3×C4#12112yes—
321C2×C3×C4#13512yes—
322C2×C3×C4AGL(1,ℤ/8)12yes—
323C2×C3×C4C22×D411yescharacteristic
324C2×C3×C4C2×#4111yescharacteristic
325C2×C3×C4C2×#4111yescharacteristic
326C2×C3×C4C2×#4111yescharacteristic
327C2×C3×C4H(5,𝔽2)11yescharacteristic
328C2×C3×C4H(5,𝔽2)11yescharacteristic
329C3×C8—22no—
330C3×C8—41no—
331C3×C8—42no—
332C24—21no—
333C24—22no—
334C2×Q8—21no—
335C2×Q8—21no—
336C2×Q8—21no—
337C2×Q8—21no—
338C2×Q8—21no—
339C2×Q8C3×#4112yes—
340C2×D4—22no—
341C2×D4—21no—
342C2×D4—21no—
343C2×D4—21no—
344C2×D4—22no—
345C2×D4—22no—
346C2×D4—22no—
347C2×D4—21no—
348C2×D4—21no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—42no—
352C2×D4—42no—
353C2×D4—42no—
354C2×D4—42no—
355C2×D4—81no—
356C2×D4—81no—
357C2×D4C2×C3×D412yes—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—21no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4C2×C3×D412yes—
366C22×C4C2×C3×D412yes—
367C22×C4C2×C3×D411yescharacteristic
368C22×C4C2×C3×D411yescharacteristic
369C22×C4C24×C311yescharacteristic
370SD16—28no—
371D8—48no—
372C2×C8—21no—
373C2×C8—21no—
374C2×C8—21no—
375C2×C8—22no—
376C2×C8—21no—
377C2×C8—22no—
378C2×C8—21no—
379C2×C8—22no—
380C2×C8C3×#4112yes—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—22no—
386C4⋊C4—21no—
387C4⋊C4—22no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—81no—
391C4⋊C4C2×C3×D412yes—
392C22⋊C4—24no—
393C22⋊C4—44no—
394C22⋊C4—44no—
395C22⋊C4—44no—
396C22⋊C4—82no—
397C22⋊C4—82no—
398C22⋊C4—82no—
399C42—22no—
400C42—22no—
401C42—44no—
402C42—42no—
403C42C2×C3×D412yes—
404C42C3×#4112yes—
405C22×C3—42no—
406C22×C3—42no—
407C22×C3—42no—
408C22×C3—81no—
409C22×C3—81no—
410C22×C3—81no—
411C22×C3—81no—
412C22×C3—82no—
413C22×C3—82no—
414C22×C3—82no—
415C22×C3—82no—
416C22×C3—82no—
417C22×C3—82no—
418C22×C3—82no—
419C22×C3C2×#12111yescharacteristic
420C22×C3#54611yescharacteristic
421C22×C3#55011yescharacteristic
422C22×C3C2×#13511yescharacteristic
423C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
424C22×C3#57611yescharacteristic
425C22×C3#57711yescharacteristic
426C3×C4—22no—
427C3×C4—42no—
428C3×C4—42no—
429C3×C4—42no—
430C3×C4—42no—
431C3×C4—82no—
432C3×C4—81no—
433C3×C4#46314yes—
434C3×C4#52511yescharacteristic
435C3×C4#53511yescharacteristic
436C3×C4#55311yescharacteristic
437C3×C4#55511yescharacteristic
438C23—22no—
439C23—41no—
440C23—41no—
441C23—41no—
442C23—41no—
443C23—41no—
444C23—41no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23C22×C3×D411yescharacteristic
453Q8—22no—
454Q8—22no—
455Q8—41no—
456Q8—41no—
457D4—22no—
458D4—22no—
459D4—42no—
460D4—42no—
461D4—41no—
462D4—42no—
463D4—41no—
464D4—42no—
465D4—84no—
466D4—84no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—48no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—42no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4C3×#12112yes—
489C2×C4C3×#12112yes—
490C2×C4C3×#13512yes—
491C2×C4C3×AGL(1,ℤ/8)12yes—
492C2×C4C22×C3×D411yescharacteristic
493C2×C4C2×C3×#4111yescharacteristic
494C2×C4C2×C3×#4111yescharacteristic
495C2×C4C2×C3×#4111yescharacteristic
496C2×C4C3×H(5,𝔽2)11yescharacteristic
497C2×C4C3×H(5,𝔽2)11yescharacteristic
498C8—22no—
499C8—42no—
500C8—41no—
501C2×C3—82no—
502C2×C3—162no—
503C2×C3#241411yescharacteristic
504C2×C3C2×#46311yescharacteristic
505C2×C3#304511yescharacteristic
506C2×C3#330511yescharacteristic
507C2×C3#333211yescharacteristic
508C2×C3#339411yescharacteristic
509C2×C3#339911yescharacteristic
510C22—42no—
511C22—42no—
512C22—42no—
513C22—81no—
514C22—81no—
515C22—81no—
516C22—81no—
517C22—82no—
518C22—82no—
519C22—82no—
520C22—82no—
521C22—82no—
522C22—82no—
523C22—82no—
524C22C2×C3×#12111yescharacteristic
525C22C3×#54611yescharacteristic
526C22C3×#55011yescharacteristic
527C22C2×C3×#13511yescharacteristic
528C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
529C22C3×#57611yescharacteristic
530C22C3×#57711yescharacteristic
531C4—22no—
532C4—42no—
533C4—42no—
534C4—42no—
535C4—42no—
536C4—82no—
537C4—81no—
538C4C3×#46314yes—
539C4C3×#52511yescharacteristic
540C4C3×#53511yescharacteristic
541C4C3×#55311yescharacteristic
542C4C3×#55511yescharacteristic
543C3#2463111yescharacteristic, direct product factor, Sylow
544C2—82no—
545C2—162no—
546C2C3×#241411yescharacteristic
547C2C2×C3×#46311yescharacteristic
548C2C3×#304511yescharacteristic
549C2C3×#330511yescharacteristic
550C2C3×#333211yescharacteristic
551C2C3×#339411yescharacteristic
552C2C3×#339911yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#459C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C3×C4×C4⋊C4 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×C3×C4×SD16C2semidirect
3. C3×#2548C2semidirect
4. C3×#2354C2semidirect
5. C3×#2054C2semidirect
6. C3×#2012C2semidirect
7. C3×#1983C2semidirect
8. C3×#1957C2semidirect
9. C3×#1939C2semidirect
10. C3×#1926C2semidirect
11. C3×#1803C2semidirect
12. #24631C3direct (N × H)
45. C2×C4×SD16C2×C3semidirect
46. #2548C2×C3semidirect
47. #2354C2×C3semidirect
48. #2054C2×C3semidirect
49. #2012C2×C3semidirect
50. #1983C2×C3semidirect
51. #1957C2×C3semidirect
52. #1939C2×C3semidirect
53. #1926C2×C3semidirect
54. #1803C2×C3semidirect
543. C3#24631direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×C3×#4592
C2×C3×C4×SD162
C3×#25482
C3×#23542
C3×#20542
C3×#20122
C3×#19832
C3×#19572
C3×#19392
C3×#19262
C3×#18032
#246313